Higher-dimensional syntax

The aim of this paper is to present the notion of higher-dimensional syntax, which is a hierarchy of languages. Each term of a n-dimensional language will be typed by terms of the underlying n ? 1-dimensional language. This is an application of the emerging higher-dimensional category notions. On the lower dimensions live notions of syntax which are already known. So dimension zero corresponds to words, dimension one corresponds to equation-free algebraic theories and dimension two corresponds to algebraic data types available in programming languages like ML and Haskell. In this paper I show that this hierarchy continues for higher dimensions and contributes new notions of syntax. As expected syntax comes with a structural induction principle which allows deenition of several operation. The operations of substitution , uniication and type inference have been investigated. Further the operation of recursive iteration schemes like generalised map and fold algorithms have been presented Weh98]. This paper concentrates on the issue of a categorical characterisation of higher dimensional syntax using notions which are partly folklore (free mon-oidal object in a monoidal category) partly are new higher-dimensional category notions. I will characterise higher dimensional syntax as a left globular set having structure of a free Lawvere theory at each dimension with a right action and a right coaction of the lower dimension language. The right action is reminiscent to the substitution property in polymorphic type theories. The right coaction is reminiscent to the minimal typing property of a polymorphic type theory.